if the absolute child function g(x) has y-intercept at (0,5), where is the y-intercept of g(1/2x)?

a. (0,2.5)
b. (0,5)
c. (0,10)

The y-intercept occurs when \( x = 0 \). Let's analyze the transformation of the function \( g(x) \) into \( g\left(\frac{1}{2}x\right) \).

1. We know that \( g(x) \) has a y-intercept at \( (0, 5) \), meaning \( g(0) = 5 \).
2. The transformed function \( g\left(\frac{1}{2}x\right) \) still needs the value at \( x = 0 \) to find the y-intercept:
   \[
   g\left(\frac{1}{2}(0)\right) = g(0) = 5.
   \]
   
Since \( g(0) = 5 \) for both \( g(x) \) and \( g\left(\frac{1}{2}x\right) \), the y-intercept of \( g\left(\frac{1}{2}x\right) \) is still \( (0, 5) \).

Thus, the correct answer is:

**b. (0, 5)**

Would you like more details or have any further questions?

### Related Questions:
1. What happens to the graph of a function when you stretch it horizontally?
2. How does a vertical stretch affect the y-intercept of a function?
3. What is the difference between horizontal and vertical transformations in functions?
4. How can we find the x-intercept of \( g\left(\frac{1}{2}x\right) \) if it exists?
5. How does the slope of a linear function change with horizontal stretching?

**Tip:** Horizontal stretches/compressions affect the x-values, but they do not change the y-intercept unless there's a vertical shift.

If the absolute child function g(x) has y-intercept at (0,5), where is the y-intercept of g(1/2x)?

Learn how to determine the y-intercept of the transformed function g(1/2x) when the original function g(x) has a known y-intercept. This solution explores function transformations, specifically horizontal stretches, and provides a step-by-step breakdown of the impact on y-intercepts. Suitable for Grades 9-12.

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